1 /* Single-precision floating point square root.
2 Copyright (C) 1997-2013 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <http://www.gnu.org/licenses/>. */
20 #include <math_private.h>
21 #include <fenv_libc.h>
27 static const float almost_half
= 0.50000006; /* 0.5 + 2^-24 */
28 static const ieee_float_shape_type a_nan
= {.word
= 0x7fc00000 };
29 static const ieee_float_shape_type a_inf
= {.word
= 0x7f800000 };
30 static const float two48
= 281474976710656.0;
31 static const float twom24
= 5.9604644775390625e-8;
32 extern const float __t_sqrt
[1024];
34 /* The method is based on a description in
35 Computation of elementary functions on the IBM RISC System/6000 processor,
36 P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
37 Basically, it consists of two interleaved Newton-Raphson approximations,
38 one to find the actual square root, and one to find its reciprocal
39 without the expense of a division operation. The tricky bit here
40 is the use of the POWER/PowerPC multiply-add operation to get the
41 required accuracy with high speed.
43 The argument reduction works by a combination of table lookup to
44 obtain the initial guesses, and some careful modification of the
45 generated guesses (which mostly runs on the integer unit, while the
46 Newton-Raphson is running on the FPU). */
49 __slow_ieee754_sqrtf (float x
)
51 const float inf
= a_inf
.value
;
57 /* Variables named starting with 's' exist in the
58 argument-reduced space, so that 2 > sx >= 0.5,
59 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
60 Variables named ending with 'i' are integer versions of
61 floating-point values. */
62 float sx
; /* The value of which we're trying to find the square
64 float sg
, g
; /* Guess of the square root of x. */
65 float sd
, d
; /* Difference between the square of the guess and x. */
66 float sy
; /* Estimate of 1/2g (overestimated by 1ulp). */
68 float e
; /* Difference between y*g and 1/2 (note that e==se). */
69 float shx
; /* == sx * fsg */
70 float fsg
; /* sg*fsg == g. */
71 fenv_t fe
; /* Saved floating-point environment (stores rounding
72 mode and whether the inexact exception is
74 uint32_t xi
, sxi
, fsgi
;
77 GET_FLOAT_WORD (xi
, x
);
78 fe
= fegetenv_register ();
80 sxi
= (xi
& 0x3fffffff) | 0x3f000000;
81 SET_FLOAT_WORD (sx
, sxi
);
82 t_sqrt
= __t_sqrt
+ (xi
>> (23 - 8 - 1) & 0x3fe);
86 /* Here we have three Newton-Raphson iterations each of a
87 division and a square root and the remainder of the
88 argument reduction, all interleaved. */
90 fsgi
= (xi
+ 0x40000000) >> 1 & 0x7f800000;
92 sg
= sy
* sd
+ sg
; /* 16-bit approximation to sqrt(sx). */
93 e
= -(sy
* sg
- almost_half
);
94 SET_FLOAT_WORD (fsg
, fsgi
);
97 if ((xi
& 0x7f800000) == 0)
100 sg
= sg
+ sy
* sd
; /* 32-bit approximation to sqrt(sx),
101 but perhaps rounded incorrectly. */
104 e
= -(sy
* sg
- almost_half
);
107 fesetenv_register (fe
);
110 /* For denormalised numbers, we normalise, calculate the
111 square root, and return an adjusted result. */
112 fesetenv_register (fe
);
113 return __slow_ieee754_sqrtf (x
* two48
) * twom24
;
118 /* For some reason, some PowerPC32 processors don't implement
120 #ifdef FE_INVALID_SQRT
121 feraiseexcept (FE_INVALID_SQRT
);
123 fenv_union_t u
= { .fenv
= fegetenv_register () };
124 if ((u
.l
[1] & FE_INVALID
) == 0)
126 feraiseexcept (FE_INVALID
);
132 #undef __ieee754_sqrtf
134 __ieee754_sqrtf (float x
)
138 /* If the CPU is 64-bit we can use the optional FP instructions. */
141 /* Volatile is required to prevent the compiler from moving the
142 fsqrt instruction above the branch. */
143 __asm
__volatile (" fsqrts %0,%1\n"
147 z
= __slow_ieee754_sqrtf (x
);
151 strong_alias (__ieee754_sqrtf
, __sqrtf_finite
)